Fungrim entry: c1bee1

\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) - \pi i

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(a\right) \lt 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(b\right) \gt 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \operatorname{Im}\!\left(b\right) - \operatorname{Re}\!\left(b\right) \operatorname{Im}\!\left(a\right) \lt 0

TeX:

\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) - \pi i

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(a\right) \lt 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(b\right) \gt 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \operatorname{Im}\!\left(b\right) - \operatorname{Re}\!\left(b\right) \operatorname{Im}\!\left(a\right) \lt 0

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log\!\left(z\right)$	Natural logarithm
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`Im`	$\operatorname{Im}\!\left(z\right)$	Imaginary part
`Re`	$\operatorname{Re}\!\left(z\right)$	Real part

Source code for this entry:

Entry(ID("c1bee1"),
    Formula(Equal(AnalyticContinuation(Log(z), z, a, b), Sub(Log(Neg(z)), Mul(ConstPi, ConstI)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Less(Im(a), 0), Greater(Im(b), 0), Less(Sub(Mul(Re(a), Im(b)), Mul(Re(b), Im(a))), 0))))

Topics using this entry

Natural logarithm